Computation of some essential dimensions

by Markus Rost (Preprint, August 2000, December 2008, 19 pages)

In these notes we show that the essential dimension of PGL₄ is equal to 5.

Along the way we discuss (in a rather unsystematic manner) generalities on essential dimension and degree formulas.

Full text (version of December 13, 2008): [pdf]

Update December 2008:

There was a gap in the proof of Lemma 11.3, which is hopefully now fixed.
The result is proved (with entirely different methods) in the preprint "Essential p-dimension of PGL(p²)" by A. Merkurjev (November 2008).

Related texts

Essential p-dimension of PGL(p²), A. Merkurjev, J. Amer. Math. Soc. 23 (2010), no. 3, 693-712. MR 2629984, Zbl 1231.20048.
Cohomological invariants of central simple algebras of degree 4, S. Baek and G. Berhuy, Arch. Math. (Basel) 98 (2012), no. 5, 415-425. MR 2922684, Zbl 1264.12004.

by Markus Rost (Notes, March 2002, 3 pages)

In this note we compute the essential dimension of a Galois module which is cyclic of order 4 (in characteristic different from 2).

Full text (version of March 2, 2002): [pdf]

Update: The result is contained (in greater detail and with reference to this preprint) as Theorem XII.37.2 in the book

An Introduction to Galois Cohomology and its Applications, G. Berhuy, London Mathematical Society Lecture Note Series, 377 (2010). MR 2723693, Zbl 1207.12003.

by Markus Rost (Notes, January 2017, 4 pages)

In this note we describe the cohomological invariants (mod 2) for a Galois module which is cyclic of order 4 (in characteristic different from 2).

Full text (version of January 15, 2017): [pdf]